Q: A triangle has sides $$a = 7$$, $$b = 10$$, and angle $$C = 45$$^{\circ}$$. Using the Law of Cosines, what is the length of side c$ opposite angle C$?

$$c = \sqrt{7^2$ + 10^2$ - 2 \times 7 \times 10 \times \cos(45$$^{\circ}$$)} \approx 7.6$$

Q: Which of the following is the correct formula for the dot product of two vectors $$\\vec{a} = \langle a_1$, a_2$ \rangle$$ and $$\\vec{b} = \langle b_1$, b_2$ \rangle$$?

$$\\vec{a} \cdot \\vec{b} = a_1b_1 + a_2b_2$$

Question 1

Which of the following best describes a function $$f(x) as $$covered in Precalculus?

Accepted Answer

A relation where each input is paired with exactly one output.

Question 2

Given the function $$f(x) = 2x^2$ - 3x + 1$$, which transformation would result in the graph $$g(x) = 2(x+2)^2$ - 3(x+2) + 1$$?

Accepted Answer

A horizontal shift left by 2 units.

Question 3

A hospital uses the exponential function $$P(t) = 500e$$^{0.05t}$$ to model the population of bacteria in a sample, where t$ is in hours. What is the population after 10 hours?

Accepted Answer

$$P(10) = 500e$$^{0.5}$$ \approx 824$$

Question 4

Which of the following is the correct definition of an inverse function $$f^{-1}(x$$)$$?

Accepted Answer

A function that undoes the operation of $$f(x)$$, so that $$f(f^{-1}(x)) = x$.

Question 5

A chemical reaction is modeled by the logarithmic function $$C(t) = rac{1}{2} \ln(t+1)$$, where $$C is $$concentration and $$t is $$time in minutes. What is the concentration at $$t = 3$$?

Accepted Answer

$$C(3) = \frac{1}{2} \ln(4) \approx 0.693$$

Question 6

Which of the following is the correct solution to the equation $$2^{x} = 16$?

Accepted Answer

$$x = 4$$

Question 7

A rational function $$f(x) = \frac{2x+3}{x-1}$$ has which of the following properties?

Accepted Answer

A vertical asymptote at $$x = 1$$

Question 8

Which of the following is the correct value of $$\sin(135$$^{\circ}$$)$$?

Accepted Answer

$$\frac{\sqrt{2}}{2}$$

Question 9

A triangle has sides $$a = 7$$, $$b = 10$$, and angle $$C = 45$$^{\circ}$$. Using the Law of Cosines, what is the length of side c$ opposite angle C$?

Accepted Answer

$$c = \sqrt{7^2$ + 10^2$ - 2 	imes 7 	imes 10 	imes \cos(45$$^{\circ}$$)} \approx 7.6$$

Question 10

Which of the following is a correct trigonometric identity?

Accepted Answer

$$\sin^2$(x) + \cos^2$(x) = 1$$

Question 11

A circle is defined by the equation $$(x-3)^2$$ + $$(y+2)^2 = 16$. What is the center and radius of the circle?

Accepted Answer

Center $$(3, -2)$$, radius $$4$$

Question 12

Which of the following is the correct binomial expansion for $$(x+2)^3$$?

Accepted Answer

$$x^3$$ + $$6x^2 + 12x + 8$

Question 13

A vector $$\vec{v}$$ has components $$\vec{v} = \langle 3, 4 angle. $$What is the magnitude of $$\vec{v}$$?

Accepted Answer

$$5$$

Question 14

Which of the following is the correct formula for the dot product of two vectors $$\vec{a} = \langle a_1$, a_2$ angle$$ and $$\vec{b} = \langle b_1$, b_2$ angle$$?

Accepted Answer

$$\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$$

Question 15

Which of the following is the correct polar form for the point $$(x, y) = (3, 3)$$?

Accepted Answer

$$(r, 	heta) = (3\sqrt{2}, 45$$^{\circ}$$)$$

Question 16

Which of the following is the correct formula for the sum of the first $$n$$ terms of an arithmetic sequence with first term $$a_1$$ and common difference $$d$$?

Accepted Answer

$$S_n = \frac{n}{2}[2a_1 + (n-1)d]$$

Question 17

A geometric sequence has first term $$a_1 = 5$ and common ratio r = 3$. What is the fifth term a_5$?

Accepted Answer

$$a_5 = 5$$ 	imes $$3^{4} = 405$

Question 18

Which of the following is the correct equation for a parabola with vertex at $$(2, -1)$$ and opening upward?

Accepted Answer

$$y = a(x-2)^2$ - 1$$

Question 19

Which of the following is the correct formula for the ellipse centered at $$(0,0)$$ with major axis along the $$x-$$axis, $$a = 5$$, $$b = 3$$?

Accepted Answer

$$\frac{x^2$}{25} + \frac{y^2$}{9} = 1$$

Precalculus MATH-2412 Syllabus and Course Structure Study Guide

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